Synthesis of Active and Passive Compatible Impedances

Abstract

This paper is concerned with the following problem: given two rational functions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z_1(s)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z_0(s)</tex> , otherwise arbitrary but for which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R + Z_1(s)</tex> has no zeros in the right-half plane, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z_1(s)</tex> is to be realized as the driving-point impedance of a lossless coupling two-port terminated in the impedance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z_0(s)</tex> . This problem had been previously considered and solved by Schoeffler and by Wohlers when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z_1(s)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z_0(s)</tex> are positive real functions and the coupling network is reciprocal. Necessary and sufficient conditions are given here for realizability in the contemplated form when neither of the two impedances are necessarily positive real and when the coupling network may be reciprocal or nonreciprocal, but still lossless. A realization procedure is described and examples are given to illustrate the approach.